Lipshitz--Sarkar stable homotopy type for certain planar trivalent graphs with perfect matchings

We develop a space-level refinement of the $2$-factor homology by constructing a stable homotopy type associated to a certain family G of planar trivalent graphs equipped with perfect matchings. Specifically, we define a cover functor from the $2$-factor flow category C(ΓM) to the cube flow category CC(n), where the perfect matching graph ΓM represents a planar trivalent graph G together with a perfect matching M, such that (G,M) ∈ G. By applying the Cohen--Jones--Segal realization to the $2$-factor flow category C(ΓM), we obtain the $2$-factor spectrum. This spectrum serves as a space-level version of the $2$-factor homology, analogous to the Lipshitz--Sarkar Khovanov spectrum for links. We show that the cohomology of the $2$-factor spectrum with Z₂-coefficients is isomorphic to the $2$-factor homology, as defined by Baldridge. We prove that the stable homotopy type of the $2$-factor spectrum is an invariant of planar trivalent graphs G equipped with perfect matchings M, whenever (G, M) ∈ G. Furthermore, we show that the closed webs obtained by performing flattenings at each crossing of an oriented link diagram in the context of sl₃ link homology belong to the family G.